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Regression Analysis

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Suppose your employer buys you a copy of Excel and you decide to learn regression analysis. You gather some historical data on Sales and Advertising expenses and run a simple regression using Sales as the dependent variable and Advertising Expense as the independent variable. The results you get are shown below, with the relevant items I want you to focus on in boldface. After pondering the results a while, you show the data to your boss and she asks the questions listed below the table. Answer the questions, keeping in mind that you are explaining your answers to a supervisor who is not conversant with economic theory. Note: the key to this question is that you are being asked to provide statistical answers, but phrasing the explanations for the statistics in a fashion that a non-statistical person can understand (a very real and common situation).

Regression Statistics
Multiple R 0.834
R Square 0.747
Adjusted R Square 0.75
Standard Error 317.105
Significance F .00001

Coefficients Standard Error t Stat P-value Lower 95%
Intercept 10,000.0 81.72330008 0.551528 0.588449 -127.3486476
X Variable 1 -5.5 0.153318278 8.4 9.61E-06 0.627810145
a. How do I convert the above information to an equation I can use? Write out that equation using the numbers provided and the following information:
Sales (S) is the dependent variable
Advertising Expense (A) is the independent variable
b. Someone who knows statistics told me that the t value (labeled t stat in boldface above) refers to the significance of the X variable. They used a table to tell me the t value of 8.4 indicates that the X variable is statistically significant at the .05 level. What does that mean in practical terms?
c. What would this firmâ??s sales be if it didnâ??t advertise? How does one interpret that number, that is, what does it represent?
d. There is something basically wrong with the regression equation (the answer to part a), i.e. the equation describes a relationship that probably isnâ??t correct. Explain what is wrong and why?

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Statistics Problems - Regression Analysis, Autocorrelation, Multicollinearity

1. Suppose an appliance manufacturer is doing a regression analysis, using quarterly time-series data, of the factors affecting its sales of appliances. A regression equation was estimated between appliance sales (in dollars) as the dependent variable and disposable personal income and new housing starts as the independent variables. The statistical tests of the model showed large t-values for both independent variables, along with a high r2 value. However, analysis of the residuals indicated that substantial autocorrelation was present.

a. What are some of the possible causes of this autocorrelation?

b. How does this autocorrelation affect the conclusions concerning the significance of the individual explanatory variables and the overall explanatory power of the regression model?

c. Given that a person uses the model for forecasting future appliance sales, how does this autocorrelation affect the accuracy of these forecasts?

d. What techniques might be used to remove this autocorrelation from the model?

2. Suppose the appliance manufacturer discussed in Exercise 1 also developed another model, again using time-series data, where appliance sales was the dependent variable and disposable personal income and retail sales of durable goods were the independent variables. Although the r2 statistic is high, the manufacturer also suspects that serious multicollinearity exists between the two independent variables.

a. In what ways does the presence of this multicollinearity affect the results of the regression analysis?

b. Under what conditions might the presence of multicollinearity cause problems in the use of this regression equation in designing a marketing plan for appliance sales?

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